The questions in 1-5 are concerned with functions of two variables. 1. Let f(x, y) = x2y + 1. Find (a) f(2, 1) (b) f(1, 3) (c) f(3a, a) (d) f(ab, a ab) (e) f(0, 0) 2. Let g(x) = x sin x. Find (a) g(x/y) (b) g(xy) (c) g(x xy) 3. Find g(u(x, y), v(x, y)) if g(x, y) = y sin(x2y), u(x, y) = x2y3 , and v(x, y) =
The questions in 1-5 are concerned with functions of two variables.
1. Let f(x, y) = x2y + 1. Find
(a) f(2, 1)
(b) f(1, 3)
(c) f(3a, a)
(d) f(ab, a ab)
(e) f(0, 0)
2. Let g(x) = x sin x. Find
(a) g(x/y)
(b) g(xy)
(c) g(x xy)
3. Find g(u(x, y), v(x, y)) if g(x, y) = y sin(x2y), u(x, y) = x2y3
, and v(x, y) =
πxy
4. Find F(g(x), h(y)) if F(x, y) = xexy
, g(x) = x3
, and h(y) = 3y + 1.
5. Let g(x, y) = ye
3x, x(t) = ln(t2 + 1), and y(t) = √
t. Find g(x(t), y(t)).
6. Suppose that the concentration C in mg/L of medication in a patient’s
bloodstream is modeled by the function C(x, t) = 0.2x(e
0.2t
eet), where
x is the dosage of the medication in mg and t is the number of hours since
the beginning of administration of the medication.
(a) Estimate the value of C(25, 3) to two decimal places. Include appropriate units and interpret your answer in a physical context.
(b) If the dosage is 100 mg, give a formula for the concentration as a
function of time t.
(c) Give a formula that describes the concentration after 1 hour in terms
of the dosage x.
The exercises in 7-10 involve functions of three variables.
7. Let f(x, y, z) = xy2z3
. Find
(a) f(2, 1, 2)
(b) f( 3, 2, 1)
(c) f(t, t2, t)
(d) f(a, a, a)
(e) f(a + b, a
b, b)
8. Let f(x, y, z) = zxy + x. Find
(a) f(x + y, x
y, x2)
(b) f(xy, y/x, xz)
9. Find g(u(x, y, z), v(x, y, z), w(x, y, z)) if g(x, y, z) = z sin xy, u(x, y, z) =
x2z3
, v(x, y, z) = πxyz, and w(x, y, z) = xy/z
10. Find F(f(x), g(y), h(z)) if F(x, y, z) = yexyz, f(x) = x2
, g(y) = y + 1 and
h(z) = z2.
In questions 11-14, sketch the domain of f. Use solid lines for
portions of the boundary included in the domain and dashed
lines for portions not included.
11. f(x, y) = ln(1
x2
y2)
12. f(x, y) = px2 + y2
4
13. f(x, y) =
1
x
y2
14. f(x, y) = ln xy
In questions 29 & 30, determine whether the statement is true
or false. Explain your answer.
29. If D is an open set in 2-space or in 3-space, then every point in D is an
interior point of D.
30. If f(x, y) →
L as (x, y)
approaches
(0, 0) along the x-axis, and if
f(x, y
) →
L as (x, y
) approaches
(0
, 0) along the y-axis, then
lim
(x,y)→(0,0)
f(x, y
) =
L.
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